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A simple intermediary orbit for the J2 problem
P. Oberti
To cite this version:
P. Oberti. A simple intermediary orbit for the J2 problem. Astronomy and Astrophysics - A&A, EDP
Sciences, 2005, 437, pp.333-338. �10.1051/0004-6361:20042406�. �hal-00419070�
DOI: 10.1051/0004-6361:20042406
c
ESO 2005
Astrophysics
&
A simple intermediary orbit for the
J
2
problem
P. Oberti
Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France e-mail:Pascal.Oberti@obs-nice.frReceived 22 November 2004/ Accepted 21 February 2005
Abstract. A simple precessing ellipse suitable for satellites with moderate eccentricities and inclinations is built in the Hamiltonian context of the J2 problem. Easy to use as a first intermediary orbit, it provides a substantially closer starting
point for series expansions than the regular Keplerian ellipse.
Key words.planets and satellites: general – celestial mechanics
1. Introduction
The J2problem deals with the motion of a satellite around an oblate planet, and an intermediary orbit is an integrable ap-proximation used as a starting point for developing the prob-lem more fully. Using the spherical coordinates r, θ, ϕ, Pr, Pθ, and Pϕdefined in the planet’s equatorial plane and defining the functions
σ = σ2
x+ σ2y+ σ2z, σx= Pϕsin θ− Pθtan ϕ cos θ,
σy= −Pϕcos θ− Pθtan ϕ sin θ, σz= Pθ, the Hamiltonian of the J2problem is
K=1 2P 2 r + 1 2 σ2 r2 − 2µ r −µJ r3 + 3µJ r3 sin 2ϕ,
where µ is the mass of the planet multiplied by the gravitational constant, and J = 1
2J2r 2
e stands for half the planet’s first zonal harmonic coefficient J2multiplied by the square of its equato-rial radius.
This is not an integrable problem (Irigoyen & Simó 1993), but it can be approximated by series expansions from suitable intermediary orbits (Deprit 1981; Floría 1993). For satellites with moderate eccentricities and inclinations, intermediary or-bits can be less sophisticated than the aforementioned ones. With K0= 1 2P 2 r + 1 2 σ2 r2 − 2µ r , J= J 3 2 σ2 z σ2 − 1 2 ,
the previous Hamiltonian reads
K= K0− µJ r3 − 3µJ r3 σ 2 x+ σ2y 2σ2 1− 2 σ2sin2ϕ σ2 x+ σ2y ,
where the first term following K0takes into account the satel-lite’s inclination, and the second term is mainly a short-period oscillating perturbation. At first sight, there are then two handy
candidates for a simple intermediary orbit, deriving from the Hamiltonian K0and the Hamiltonian
K1= 1 2P 2 r+ 1 2 σ2 r2 − 2µ r −µJ r3·
Both are integrable problems. The first one (the 2-Body prob-lem) is easy to solve but leads to rather long series expansions to reach K. The second one provides a much closer starting point but involves the somewhat less convenient elliptic func-tions in the solution. The purpose of this paper is to build an easy-to-use intermediary orbit between K0 and K1suitable for satellites with moderate eccentricities and inclinations. 2.K0 and the Keplerian ellipse
Some classical results are useful for the following sections. The orbit deriving from K0 can be defined by the osculat-ing elements computed from the prime integrals of the motion
σx, σy, σz, and K0= k0(Duriez 1989). When−µ2/2σ2 < k0 < 0, there are two different positive values of r for which Pr= 0. The motion is bounded and the orbit is the Keplerian ellipse defined by a, e, i,Ω, ω, and τ: a= − µ 2k0 , e= 1+ 2k0 σ2 µ2, sin i= σ2 x+ σ2y σ , cos i=σz σ, sinΩ = σx σ2 x+ σ2y , cosΩ = −σy σ2 x+ σ2y if sin i 0. The value of τ is computed from the initial values of the canonical variables at t= t0:
e sin u= √µa, ecosu = 1 −rPr r a,
τ = (e sin u − u)
a3 µ +t0,
334 P. Oberti: A simple intermediary orbit for the J2problem
and the value of ω can be computed from sin v= √ 1− e2sin u 1− e cos u , cos v= cos u− e 1− e cos u; sin i= 0: v + ω = θ; sin i 0: sin (v+ ω) = σ sin ϕ σ2 x+ σ2y , cos (v+ ω) = Pϕcos ϕ σ2 x+ σ2y ·
The variations of r, θ, and ϕ as functions of the time are
u− e sin u =
µ
a3(t− τ) , r = a (1 − e cos u) , sin ϕ= sin i sin (v + ω) ,
cos ϕ sin θ= cos (v + ω) sin Ω + sin (v + ω) cos Ω cos i, cos ϕ cos θ= cos (v + ω) cos Ω − sin (v + ω) sin Ω cos i. The canonical Delaunay transformation
= µ a3(t− τ) , L = √µa, g = ω, G = µa 1 − e2= σ, h= Ω, H = µa 1 − e2cos i= σ z turns the Hamiltonian of the J2problem into
K= − µ 2 2L2 − µJ r3 − 3µJ r3 sin2i 2 cos 2 (v+ ω) ,
where J = J1−32sin2i, and where non-canonical variables have to be expressed in canonical ones.
3. The dressing ofK1
The functions σxand σyare no longer prime integrals of the differential system derived from K1, but σ2x+σ2yand σzare still prime integrals of this system, together with K1 = k1. Finding different real values of r for which Pr= 0 requires
108µ2J2k21+ 2σ218µ2J− σ4k1+ µ2 16µ2J− σ4< 0, leading to 12µ2J σ4 < 1, R = 1−12µ 2J σ4 , D 3= 2µ J, σ6−1 + 3R2− 2R3 54D6 < k1 < σ6−1 + 3R2+ 2R3 54D6 ·
The lower bound of k1 is always negative. Raising the energy level of a Hamiltonian in order that the lower bound of the motion range equals 0 generally simplifies the Hamiltonian’s expression: K0+ µ2 2σ2 = 1 2P 2 r+ 1 2 µ σ − σ r 2 ·
Let’s try the process on K1:
2 × K1− 1 2P 2 r+ σ61− 3R2+ 2R3 54D6 = σ 61− 3R2+ 2R3 27D6 − 2µ r + σ2 r2 − 2µ J r3 = σ6 27D6 − σ4 3D3r+ σ2 r2 − D3 r3 −3σ6R2 27D6 1−2R 3 + σ4 3D3r 1 − 12µ2J σ4 = σ2 3D2 − D r 3 −3 σ2 3D2 − D r σ2 R 3D2 2 + 2 σ2 R 3D2 3 = σ2 (1+ 2R) 3D2 − D r σ2 (1− R) 3D2 − D r 2 = 1 +2R3 −2µ σ2J r µ σ 2 1+ R −σ r 2 = 1+ 2R 3 σ2 r2 − 2µ r 2 1+ R −2µ J r3 × µr σ2 2 1+ R − 1 2 +σµ22 1+ 2R 3 2 1+ R 2 ·
With the functions
σ = σ 1+ 2R 3 , µ = 2 3µ 1+ 2R 1+ R ,
and the relation
σ61− 3R2+ 2R3 54D6 = µ2 2σ2 1+ 2R 3 2 1+ R 2 ,
the Hamiltonian’s expression is
K1= 1 2P 2 r+ 1 2 σ2 r2 − 2µ r −µJ r3 µr σ2 − 1 2 ·
4. ˜K0and the Hamiltonian ellipse
Let’s define the Hamiltonian
K0= 1 2P 2 r+ 1 2 σ2 r2 − 2µ r = k0.
The first step for finding the solution is similar to solving the 2-Body problem. When −µ2/2σ2 < k0 < 0, the motion is bounded: a= − µ 2k0 , e= 1+ 2k0 σ2 µ2, a (1− e) ≤ r ≤ a (1 + e) .
The canonical action-variable (Henrard 1989) I = 1 2π Prdr = 1π a(1+e) a(1−e) 2k0− σ2 r2 + 2µ r 1 2 dr= µ −2k0 − σ
turns the Hamiltonian into
K0= − µ 2 2 (I+ σ)2,
and the computation of the canonical angle-variable
ψ = ∂K0 ∂I r a(1−e) ∂Pr ∂k0 dρ = µ2 (I+ σ)3 r a(1−e) 2k0− σ2 ρ2 + 2µ ρ −1 2 dρ = arccosa− r ae − e sin arccos a− r ae leads to dψ dt = ∂K0 ∂I = µ a3, r= a (1 − e cos u) , u − e sin u = µ a3(t− τ) ,
where the value of τ is computed from the initial values of the canonical variables at t= t0: e sin u= rPr µa, e cos u= 1 − r a, τ = (e sin u − u) a3 µ + t0.
The differential system derived from K0is dr dt = Pr, dθ dt = C1Pθ r2cos2ϕ + C2 r2, dϕ dt = C1Pϕ r2 , dPr dt = σ2 r3 − µ r2, dPθ dt = 0, dPϕ dt = − C1P2θsin ϕ r2cos3ϕ , where C1 = σ σ ∂σ ∂σ − r σ ∂µ ∂σ, C2= σ ∂ σ ∂σz − r ∂µ ∂σz , ∂R ∂σ = 12µ2J Rσ5 9 2 σ2 z σ2 − 1 , ∂R ∂σz = − 18µ2Jσ z Rσ6 , ∂σ ∂σ = 1+ 2R 3 + σ 3 ∂R ∂σ 3 1+ 2R, ∂σ ∂σz = σ 3 ∂R ∂σz 3 1+ 2R, ∂µ ∂σ = 2 3 ∂R ∂σ µ (1+ R)2, ∂µ ∂σz = 2 3 ∂R ∂σz µ (1+ R)2·
Defining the inclination by
sin i= σ2 x+ σ2y σ , cos i= σz σ,
and using the relation dt=
a3
µ (1− e cos u) du
turn the equation for ϕ into cos ϕ sin2i− sin2ϕ dϕ= ∂∂σσ √ 1− e2 1− e cos u− a µ ∂µ ∂σ du.
The variable ϕ is not periodic. Let’s use the relation arcsin sin ϕ sin i = arctan σ sin ϕ Pϕcos ϕ ,
and extend from−π to π the value range of the right-hand func-tion according to the signs of sin ϕ and Pϕ. Then, if m is the largest integer≤ µ/a3(t− τ) /2π, n is a suitable integer, and v is defined by sin v= √ 1− e2sin u 1− e cos u , cos v= cos u− e 1− e cos u, where 0≤ v < 2π, the solution is
2nπ + arctan σ sin ϕ Pϕcos ϕ = ∂∂σσ(2mπ+ v) − a µ ∂µ ∂σu+ ω = v + ∂σ ∂σ −1 (2mπ+ v) − a µ ∂µ ∂σu+ ω + 2mπ = v + ∂σ ∂σ −1 (2mπ+ v − u) + ∂σ ∂σ −1− a µ ∂µ ∂σ e sin u +ω + ∂σ ∂σ −1− a µ ∂µ ∂σ µ a3(t− τ) + 2mπ. The variable r is periodic of period 2π a3/µ. The value of u can then be computed from
u− e sin u =
µ
a3(t− τ) − 2mπ,
which means that 0 ≤ u < 2π. This convention turns u into 2mπ+ u in the previous solution. Thus, defining
v = v + ∂σ ∂σ −1 (v − u) + ∂σ ∂σ −1− a µ ∂µ ∂σ e sin u, ω = ω + ∂σ ∂σ −1− a µ ∂µ ∂σ µ a3(t− τ) ,
336 P. Oberti: A simple intermediary orbit for the J2problem
where 0≤ u, v < 2π, the value of ω can be computed from sin i= 0: v+ ω = θ; sin i 0: sin (v+ ω) = σ sin ϕ σ2 x+ σ2y , cos (v+ ω) = Pϕcos ϕ σ2 x+ σ2y ,
and the variations of ϕ are given by sin ϕ= sin i sin (v+ ω) .
The anglevperiodically oscillates around v, and ω is a precess-ing angle. They generalize v and ω when J 0. The variations of θ involve them, too. From the differential system derived from K0, the equation for θ is
dθ = cos i cos ϕ sin2i− sin2ϕ dϕ + ∂σ∂σ z √ 1− e2 1− e cos u− a µ ∂µ ∂σz du.
The variable θ is not periodic. Let’s use the relation arcsin tan ϕ tan i = arctan σ zsin ϕ Pϕcos ϕ ,
and extend from−π to π the value range of the right-hand func-tion according to the signs of sin ϕ and Pϕ. Then, if n is a suit-able integer, and 0≤ u, v < 2π, the solution is
θ = 2nπ + arctan σ zsin ϕ Pϕcos ϕ +∂σ∂σ z (v − u) + ∂σ ∂σz − a µ ∂µ ∂σz e sin u +Ω + ∂σ ∂σz − a µ ∂µ ∂σz µ a3(t− τ) . Thus, defining θ = θ − ∂σ∂σ z (v − u) − ∂σ ∂σz − a µ ∂µ ∂σz e sin u, Ω = Ω + ∂σ ∂σz − a µ ∂µ ∂σz µ a3(t− τ) , the value ofΩ can be computed from sinθ− Ω= σztan ϕ σ2 x+ σ2y , cosθ− Ω= Pϕ σ2 x+ σ2y if sin i 0, and the variations of θ come from
cos ϕ sinθ = cos (v+ ω) sin Ω + sin (v+ ω) cos Ω cos i, cos ϕ cosθ = cos (v+ ω) cos Ω − sin (v+ ω) sin Ω cos i. The angle θ periodically oscillates around θ, and Ω is a precess-ing angle. They generalize θ andΩ when J 0. For initial val-ues matching σ6−6µ2J3σ2
z− σ2
> 0 and −µ2/2σ2< k 0< 0, the solution of K0 is a precessing ellipse with the same fixed inclination as the Keplerian ellipse. It can be described from the osculating elements a, e, i,Ω, ω, and τ defining what can be called the Hamiltonian ellipse related to the J2problem.
5. Canonical variables
The expression of K0as a function of I and ψ implies ∂K0 ∂I = dψ dt, ∂K0 ∂σ = d dt(ω + ψ) , ∂K0 ∂σz =dΩ dt· The transformation 2= ψ = µ a3(t− τ) , L2= I + σ = µa − σ + σ, g2= (ω + ψ) − ψ = ω, G2= σ, h2= Ω, H2= σz is then canonical and turns the Hamiltonian into
K0= − µ 2 2 (L2− G2+ σ)2
·
In order to apply the Lie algorithm to a Hamiltonian (Deprit 1969), it is useful to reduce the number of variables in the in-termediary orbit. The canonical transformation
1 = 2, L1= L2− G2+ σ = µa, g1 = g2− ∂σ ∂G2 − 1 2= ω − ∂σ ∂σ −1 µ a3(t− τ) = ω − a µ ∂µ ∂σ µ a3(t− τ) , G1= G2, h1 = h2− ∂ σ ∂H2 2= Ω − ∂ σ ∂σz µ a3(t− τ) = Ω − a µ ∂µ ∂σz µ a3(t− τ) , H1= H2 begins the process:
K0= −µ 2 2L2 1 .
The Hamiltonian still depends on G1and H1because ofµ. The canonical transformation =µµ1, L= µ µL1, g = g1+ L1 µ ∂µ ∂G1 1= ω, G = G1, h= h1+ L1 µ ∂µ ∂H1 1= Ω, H = H1 ends the process:
K0= − µ2 2L2.
Then, with the relation
E= µr σ2 − 1 = e 1− e e− cos u 1+ e ,
Table 1. Standard deviations for r, θ, and ϕ over one period of r.
J e, ek, eh Keplerian ellipse: sr, sθ, sϕ Hamiltonian ellipse: sr, sθ, sϕ
10−5 0.1 0.10 0.10 0.3 0.30 0.30 0.5 0.50 0.50 5.21× 10−5 7.73× 10−4 1.54× 10−4 3.88× 10−5 7.42× 10−4 1.52× 10−4 3.36× 10−5 9.53× 10−4 1.84× 10−4 5.21× 10−6 2.33× 10−5 6.11× 10−6 1.53× 10−5 8.72× 10−5 6.65× 10−6 2.42× 10−5 2.94× 10−4 3.29× 10−5 10−4 0.1 0.10 0.10 0.3 0.30 0.30 0.5 0.50 0.50 5.21× 10−4 7.74× 10−3 1.54× 10−3 3.88× 10−4 7.44× 10−3 1.51× 10−3 3.36× 10−4 9.58× 10−3 1.84× 10−3 5.26× 10−5 2.35× 10−4 6.15× 10−5 1.54× 10−4 8.81× 10−4 6.72× 10−5 2.43× 10−4 2.97× 10−3 3.32× 10−4 10−3 0.1 0.10 0.11 0.3 0.30 0.31 0.5 0.50 0.51 5.20× 10−3 7.88× 10−2 1.55× 10−2 3.90× 10−3 7.64× 10−2 1.51× 10−2 3.38× 10−3 1.00× 10−1 1.85× 10−2 5.82× 10−4 2.56× 10−3 6.48× 10−4 1.61× 10−3 9.70× 10−3 8.01× 10−4 2.51× 10−3 3.34× 10−2 4.01× 10−3 10−2 0.1 0.10 0.22 0.3 0.30 0.41 0.5 0.50 0.64 5.17× 10−2 9.91× 10−1 1.79× 10−1 4.11× 10−2 1.09 2.07× 10−1 4.46× 10−2 3.04 1.21× 10−1 1.34× 10−2 8.80× 10−2 1.62× 10−2 2.56× 10−2 0.32 6.55× 10−2 5.29× 10−2 3.00 1.16× 10−1
Table 2. Standard deviations for r, θ, and ϕ over five periods of r.
J e, ek, eh Keplerian ellipse: sr, sθ, sϕ Hamiltonian ellipse: sr, sθ, sϕ
10−5 0.1 0.10 0.10 0.3 0.30 0.30 0.5 0.50 0.50 3.84× 10−5 1.49× 10−4 1.53× 10−4 1.11× 10−4 5.09× 10−4 2.04× 10−4 1.97× 10−4 1.19× 10−3 3.45× 10−4 1.23× 10−5 5.06× 10−5 6.06× 10−6 4.33× 10−5 2.04× 10−4 2.79× 10−5 1.02× 10−4 6.43× 10−4 1.00× 10−4 10−4 0.1 0.10 0.10 0.3 0.30 0.30 0.5 0.50 0.50 3.87× 10−4 1.51× 10−3 1.52× 10−3 1.11× 10−3 5.12× 10−3 2.05× 10−3 1.97× 10−3 1.19× 10−2 3.51× 10−3 1.24× 10−4 5.13× 10−4 6.20× 10−5 4.35× 10−4 2.06× 10−3 2.84× 10−4 1.02× 10−3 6.50× 10−3 1.04× 10−3 10−3 0.1 0.11 0.11 0.3 0.31 0.31 0.5 0.52 0.51 4.23× 10−3 1.67× 10−2 1.54× 10−2 1.15× 10−2 5.34× 10−2 2.25× 10−2 2.00× 10−2 1.24× 10−1 4.53× 10−2 1.38× 10−3 5.82× 10−3 8.40× 10−4 4.58× 10−3 2.22× 10−2 3.93× 10−3 1.07× 10−2 7.09× 10−2 1.70× 10−2 10−2 0.1 0.14 0.19 0.3 0.29 0.32 0.5 0.39 0.64 1.03× 10−1 4.58 2.40× 10−1 7.83× 10−2 4.34 0.48 0.11 13.79 0.28 3.20× 10−2 0.20 5.02× 10−2 2.92× 10−2 0.10 0.46 0.21 14.00 0.19
the Hamiltonian of the J2problem turns into
K= − µ 2 2L2 − µJ r3 E 2−3µJ r3 sin2i 2 cos 2 (v+ ω) ,
where J = J1−32sin2i, and where non-canonical variables have to be expressed in canonical ones. Building an improved intermediary orbit requires that the second term of K be smaller than the one obtained from the classical process. Then, with
|e − cos u| ≤ 1 + e, K0offers a better intermediary orbit than K0 when e < 1
2. And the smaller the better. . .
6. Numerical simulations
Two sets of numerical simulations are performed in dimension-less units with four different values of J and µ = 1. In this case, the value of J would be around 1.23× 10−5and 7.96× 10−4for the Earth and Saturn, respectively. The simulations start with initial values computed from a= 0.5, three different values of
e (0.1, 0.3, 0.5), and i= 0.2, where a, e, and i are the osculating
elements of the J2problem at the starting time. The first set of simulations compares K0and K0in their roles as first interme-diary orbits to be used as a starting point for developments. The
discrepancies between each approximation and the J2problem are computed for r, θ, and ϕ during one period of the variable
r of the considered approximation. The actual eccentricities ek (Keplerian ellipse) and eh (Hamiltonian ellipse) and the stan-dard deviations sr, sθ, and sϕfor each approximation are sum-marized in Table 1. The second set of simulations compares K0 and K0 in their roles as approximate solutions to be used for predictions over short periods of time. The discrepancies be-tween each approximation and the J2 problem are computed for r, θ, and ϕ during five periods of the variable r of the con-sidered approximation, but the constants of each approximation are fitted by least squares to the J2problem. The fitted eccen-tricities ekand ehand the standard deviations sr, sθ, and sϕfor each approximation are summarized in Table 2. In both cases,
K0 is a better option for moderate eccentricities, leading up to ten times smaller values, and a rather worse option for eh> 1
2, as expected from the theory.
7. Perturbation
The solution of K is obtained from the perturbation of the canonical Delaunay-like variables , g, h, L, G, and H involved
338 P. Oberti: A simple intermediary orbit for the J2problem
in the solution of K0. With these variables, the functions re-quired for computing r, θ, and ϕ read
J= J 3 2 H2 G2 − 1 2 , R= 1−12µ 2J G4 , ∂R ∂G = 12µ2J R G5 9 2 H2 G2 − 1 , ∂R ∂H = − 18µ2JH R G6 , σ = G 1+ 2R 3 , µ = 2 3µ 1+ 2R 1+ R , ∂σ ∂G = 1+ 2R 3 + G 3 ∂R ∂G 3 1+ 2R, ∂σ ∂H = G 3 ∂R ∂H 3 1+ 2R, ∂µ ∂G = 2 3 ∂R ∂G µ (1+ R)2, ∂µ ∂H = 2 3 ∂R ∂H µ (1+ R)2, a= µµ2L2, e= 1−µ 2 µ2 σ2 L2, sin i= 1−H 2 G2, cos i= H G, u− e sin u = µ µ, 0≤ u < 2π, sin v= √ 1− e2sin u 1− e cos u , cos v= cos u− e 1− e cos u, 0≤ v < 2π, v= v + ∂σ ∂G −1 (v − u) + ∂σ ∂G −1− L µ ∂µ ∂G e sin u, ω = g + ∂σ ∂G −1− L µ ∂µ ∂G µ µ, θ = θ − ∂H∂σ (v − u) − ∂σ ∂H − L µ ∂µ ∂H e sin u, Ω = h + ∂σ ∂H − L µ ∂µ ∂H µ µ.
The main difference with the classical case starting from K0 is that the expansion of K involves the new functions σ,µ, and their derivatives with respect to G and H. They only de-pend on these canonical variables but have rather simple ex-pressions using G, H, and R. For automatic algebraic computa-tions (Moons 1991), the non-canonical variable S = √R can be
added to the set of variables used to expand the Hamiltonian, provided some minor modifications are made in the derivative algorithms. Then, for the commonly low values of J, the new functions can be quickly expanded in powers of1− S2/S2. For very small values of e or i, non-singular canonical vari-ables can be defined in the very same way they are defined for the classical case.
8. Conclusion
The approximation K0provides a simple precessing ellipse that is easy to use as a first intermediary orbit in the Hamiltonian context of the J2problem, allowing the powerful process of Lie series expansions to compute the complete solution. Suitable for satellites with moderate eccentricities and inclinations, it can be used in the theories of many natural satellites or for non-geosynchronous artificial satellites near the Earth’s equator.
References
Deprit, A., 1969, Celest. Mech., 1, 12 Deprit, A., 1981, Celest. Mech., 24, 111
Duriez, L., 1989, Modern Methods in Celestial Mechanics, Goutelas (France: Éditions Frontières)
Floría, L., 1993, Celest. Mech. Dyn. Astr., 57, 203
Henrard, J., 1989, Modern Methods in Celestial Mechanics, Goutelas (France: Éditions Frontières)
Irigoyen, M., & Simó, C., 1993, Celest. Mech., 55, 281
Moons, M., 1991, Dept. of Math., Facultés Universitaires Notre Dame de la Paix, Namur, Belgium
